Problem: How many two-digit numbers have digits whose sum is a perfect square?
The sum of the digits of a two-digit number is at most $9+9=18.$ This means the only possible perfect square sums are $1,$ $4,$ $9,$ and $16.$ Each square has the following two-digit possibilities:

$\bullet$ $1:$ $10$

$\bullet$ $4:$ $40,$ $31,$ $22,$ $13$

$\bullet$ $9:$ $90,$ $81,$ $72,$ $63,$ $54,$ $45,$ $36,$ $27,$ $18$

$\bullet$ $16:$ $97,$ $88,$ $79$

There are $\boxed{17}$ two-digit numbers in all.